2008 / September Volume 3 No.3
On the law of the iterated logarithm for L-statistics without variance
| Published Date |
2008 / September
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|---|---|
| Title | On the law of the iterated logarithm for L-statistics without variance |
| Author | |
| Keyword | |
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| Pagination | 417-432 |
| Abstract | Let {$X,X_n; n \ge 1$} be a sequence of i.i.d. random variables with distribution function $F(x)$. For each positive integer
$n$, let
$X_{1:n} \le X_{2:n} \le \dots \le X_{n:n}$ be the order statistics of
$X_1, X_2, \dots, X_n$. Let $H(\cdot)$
be a real Borel-measurable function defined on $\mathcal{R}$ such that
$\mathbb{E}|H(X)| < \infty$
and let $J(\cdot)$ be a Lipschitz function of order one defined on $[0, 1]$. Write
$\mu = \mu(F, J, H) = \mathbb{E}(J(U)H(F_\gets (U)))$ and
$\mathbb{L}_n(F, J, H) = \frac 1n \sum^n_{i=1} J(\frac in)H(X_{i:n}), n \ge 1,$ where $U$ is a random variable with the uniform (0,1) distribution and
$F^\gets(t) = \inf \{ x; F(x) \ge t \}$,
$0 \lt t \lt 1$.
In this note, the Chung-Smirnov LIL for empirical processes and the Einmahl-Li LIL for partial sums of i.i.d. random variables without variance are used to establish necessary and sufficient conditions for having with probability 1: 0 < lim
$\sup_{n \to \infty} \sqrt{n/ \varphi(n)} | \mathbb{L}_n(F, J, H) - \mu | < \infty$,
where $\varphi(\cdot)$ is from a suitable subclass of the positive, non-decreasing, and slowly varying functions defined on $[0, \infty)$. The almost sure value of the limsup is identified under suitable conditions. Specializing our result to
$\varphi(x) = 2(\log \log x)^p , p>1$ and to $\varphi(x) = 2(\log x)\gamma, \gamma > 0$, we obtain an analog of the Hartman-Wintner-Strassen LIL for L-statistics in the infinite variance case. A stability result for L-statistics in the infinite variance case is also obtained.
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| AMS Subject Classification |
60F15, 62G30
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| Received |
2008-01-08
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| Accepted |
2008-01-09
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